Rankin-Selberg integrals in many complex variables and Rankin-Selberg integrals associated to non-unique models

Abstract

In this thesis we explore Rankin-Selberg integrals in many complex variables and Rankin-Selberg integrals associated to non-unique models. In particular we try to bring to light a possible connection between them. First, we reinterpret the integral of Kohnen-Skoruppa for the Spin L-function on GSp(4) as an integral associated to a non-unique model, following Piatetski-Shapiro and Rallis. Then we give two integrals on the quasisplit unitary group GU(2,2) in four variables. The first integral is in two complex variables, and applies to generic cusp forms on GU(2,2); it represents the product of the standard (8 dimensional representation) and the exterior square (6 dimensional representation) of the dual group. The second integral on GU(2,2) represents the exterior square L-function and is associated to a non-unique model; this integral is a reinterpretation of an integral of Gritsenko who considered holomorphic Hermitian modular forms of genus two. We then give an integral representation for the Spin L-function on GSp(6), again associated to a non-unique model. As we explain below, these non-unique model integrals on GU(2,2) and GSp(6) may be considered analogues of the integral of Kohnen-Skoruppa. Some of the results in this thesis stem from work done jointly with Shrenik Shah.